Spectral Analysis of Julia Sets

Thesis by

St anislav K. Smirnov

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

1996

(Submitted May 10, 1996)

Typeset by AMTEX

Acknowledgements

I am deeply grateful to my advisor, Prof. N. G. Makarov for his sup-

port and supervision, without which this research would not have been

completed.

It is a great pleasure to thank my advisor at St.-Petersburg State Uni-

versity Prof. V. P. Havin, who introduced me to analysis, and taught

me many things; and Prof. D. Sullivan, from whom I have learned a

lot, sometimes indirectly.

I am indebted to the California Institute of Technology and everyone

in the Mathematics Department, especially M. D'Elia, for the wonder-

ful time spent here. I wish to thank my friends I. Binder, A. Epstein,

J . Graczyk, and A. Poltoratski for many hours of interesting discus-

sions.

I am also grateful to V. Baladi, G. Jones, M. Lyubich, F. Przytycki,

D. Ruelle, M. Urbariski, A. Volberg, and many others for valuable

conversations and suggest ions.

This work was completed with a very much appreciated support by an

Alfred P. Sloan Foundat ion Doctoral Fellowship.

Abstract

We investigate different measures defined geometrically or dynamically

on polynomial Julia sets and their scaling properties. Our main concern

is the relationship between harmonic and Hausdorff measures.

We prove that the fine structure of harmonic measure at the more

exposed points of an arbitrary polynomial Julia set is regular, and di-

mension spectra or pressure for the corresponding (negative) values of

parameter are real-analytic. However, there is a precisely described

class of polynomials , where a set of preperiodic critical points can gen-

erate a unique very exposed tip, which manifests in the phase transition

for some kinds of spectra.

For parabolic and subhyperbolic polynomials, and also semihyperbolic

quadratics we analyze the spectra for the positive values of parameter,

establishing the extent of their regularity.

Results are proved through spectral analysis of the transfer (Perron-

F'robenius-Ruelle) operat or.

Contents

. Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . 1

1 . Overview and general discussion . . . . . . . . . . . . . . . 3

l a . Measures and transfer operator . . . . . . . . . . . . . . 4

l b . Fine structure of measures . . . . . . . . . . . . . . . . 20

2 . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

. Chapter 2 Negative spectra . . . . . . . . . . . . . . . . 45

1 . Quasicompactness . . . . . . . . . . . . . . . . . . . . . . . . 47

2 . Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 . Transfer operator on Sobolev spaces . . . . . . . . . . . . . 78

. Chapter 3 Parabolic case . . . . . . . . . . . . . . . . . . 84

1 . Transfer operator . . . . . . . . . . . . . . . . . . . . . . . . 85

2 . Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Chapter 4 . Semihyperbolic case . . . . . . . . . . . . . . 102

1 . Yoccoz puzzle and Hofbauer tower . . . . . . . . . . . . . . 103

2 . Analysis of the transfer operator . . . . . . . . . . . . . . . 108

3 . Analyticity of spectrum

. . . . . . . . . . . . . . . . . . . . . . . . . . . . References

List of Figures

Figure 1. Spectra of hyperbolic polynomials . . . . . . . . . . 29

Figure 2. Phase transition in negative spectra . . . . . . . . . 35

2 Figure 3. Spectra of parabolics with HDim JF > 2 - w . . 37

2 Figure 3. Spectra of parabolics with HDim JF 5 2 - j;TI . . 37

Chapter 1

Introduction

The main topic of this dissertation lies in the study of the dynamics

of polynomial iterations and the geometry of corresponding Julia sets.

We investigate different measures defined geometrically or dynamically

on Julia sets and their scaling properties. This is done via describing

a collection of characteristics, which we will call spectra.

Holomorphic dynamics. Considering a holomorphic dynamical sys-

tem, such as Julia set, one notices that dynamics yields self-similarity.

Hence geometry is "homogeneous" in different places, and the logical

way to describe it is to count "how often" certain behavior occurs in

different scales. It is also logical to expect that all geometrical objects

have also dynamical/ergodic meaning, and the same set of parameters

will describe both geometry and dynamical properties.

As usual in dynamics, one finds that hyperbolic Julia sets with ex-

panding dynamics are easier to study and have very nice properties.

CHAPTER 1. INTRODUCTION 2

Non-hyperbolic dynamics is more difficult to understand, but still some

remnants of expanding can be observed, hence one can expect to find

difficult, but interesting behavior.

Complex analysis. Approaching the problem of describing the geom-

etry of domains and their boundaries in the complex plane, one notices

that many questions can be reduced to understanding the structure of

harmonic measure. However, harmonic measure will have good scaling

properties only for self-similar sets, so one can obtain more interesting

results in such a case. On the other hand, the extremal behavior of

harmonic measure can be approximated on self-similar fractals, which

also motivates studying of their properties.

1. Overview and general discussion Julia sets represent a class of dynamical systems which are defined

seemingly easily but can exhibit very difficult properties. They were

studied, starting with the works of L. Bottcher, P. Fatou, and G. Julia

throughout this century, and very intensively over the last two decades.

Partially this interest was evoked by beautiful computer pictures, which

showed difficult "fractal" structure and resemblance to many physical

phenomena arising in nature.

For a rational function F one can define the Julia set JF c as the

complement to the set of points in whose neighborhoods iterates of F

form a normal family:

JF := \ { z : 3 U 3 z , { F n 1 v) is normal family) ,

another possible definition is the closure of all repelling periodic points:

JF := Clos { z : F n ( z ) = z , I(Fn)'(z)I > 1 for some n ) .

We mainly will be interested in the case when F is a polynomial, then

JF coincides with the boundary of domain of attraction to infinity

A(oo) := { z : F n ( z ) + oo as n + 00) ,

which is fully invariant under the action of F. The nicest class of Julia

sets are the so called hyperbolic ones, for which dynamics F on the

Julia set is expanding, i.e. ((Fn)'(z) 1 > CQn, C > 0, Q > 1 for any

z E JF and n E Z+. For these definitions and basic properties of the

Julia sets see monograph [DH] of A. Douady and J. Hubbard; books

[Be] by A. Beardon, [Mil] by J. Milnor and [CG] by L. Carleson

and T. Gamelin, the latter follows a more analytical approach. The

expository paper [ELy] of A. Eremenko and M. Lyubich gives a good

present ation of their ergodic properties.

la. Measures and transfer operator Trying to understand the properties of a Julia set, one can start look-

ing at the different measures (defined geometrically or dynamically),

and at their behavior under dynamics.

1.1. Entropy and Balanced Measures

Balanced measures. A first logical class of measures to look at are

the invariant ones. However, this class is too broad; so we can start

with considering the so called balanced measures, where the mass is

uniformly distributed among the preimages. Namely

deg F . p (A) = p (F (A)) , if F is injective on A ,

i.e. Jacobian of p is equal to deg F. H. Brolin in [Bro] has proved

existence of a balanced measure for polynomials and has shown that

it coincides with the equilibrium (harmonic) measure from potential

theory, which is well-defined since capacity of the Julia set is equal to

1. He has also shown that it has strong mixing property, namely

Uniqueness of the balanced measure was established later by A. F'rei-

re, A. Lopes, and R. Maii6 in [FLM], [Maiil]; also in [BGH] by

M. F. Barnsley, J. S. Geronimo, and A. N. Harrington. See the latter

paper, [BH], and [Lo21 for the properties of potential, generated by

the balanced (equilibrium) measure and its connections to Pade ap-

proximations and theory of moments.

Another way to view the balanced measure is to notice that preim-

ages F-"z as n tends to infinity have uniform distribution with respect

to it, hence one can construct the balanced measure as a weak limit of

the sums

w - lim d-" 6, . n-oo

Here 6, denotes the 6-measure supported at point y. This process can

be viewed as considering operator L* acting on measures:

and analyzing its iterates. In such a way M. Lyubich has constructed

balanced measures for rational functions (see [Ly2-31).

One can play with these sums (or operator) in a different way,

putting a unit mass at some point, l /d masses at its preimages, l /d2

at their preimages, etc., and taking a weak limit of measures obtained

at the n-th step, properly normalized. A similar approach to another

measure will appear to be useful later.

Harmonic measure and symbolic dynamics. One would like to

have a good model of the dynamics on the Julia set. If it is connected

and locally connected, then (via the Riemann uniformization map) dy-

namics z H zd (d := deg F) o n the unit circle gives us a topological

model (modulo some lamination). Hence it is logical to expect that it

will also be a proper metric model.

A more sophisticated way, which also works in a non-connected case,

is symbolic dynamics, which models the dynamics F on the Julia set by

equal-weighted one-sided Bernoulli shift on Zr. It was first introduced

for Julia sets by M. Jakobson and J. Guckenheimer (see [Jl-41 and

[Gu]) and used extensively later. Bernoulli shift indeed appears to

be a proper metric model: R. Maii6 has shown in [Mafi2] that for

some iterate N, (FN , p) is equivalent to the equal-weighted one-sided

Bernoulli shift on dN symbols.

Length plays an important role in the dynamics on the unit circle,

particularly it is balanced and maximizes the entropy (with a value of

log d). The same is true for its analogue - d-adic measure on ZT.

In the connected case mapping the length from the unit circle we ob-

tain harmonic measure on the Julia set. As was mentioned, H. Brolin

proved that it coincides with the balanced measure, moreover A. Lopes

has shown that this property characterizes the polynomials among the

rational functions (see [Loll and also paper of M. Lyubich and A. Vol-

berg [LVl-21). Harmonic measure is a very important tool in under-

standing the structure of some set. Hence the unique balanced measure,

except for dynamical, plays quite an important geometric role.

Harmonic measure also admits a probabilistic approach, since it mea-

sures the probability of a set being hit by a Brownian motion, which

is invariant under the conformal maps. Hence it naturally fits in the

framework of holomorphic dynamics. This connection of probability to

dynamics hasn't remained unnoticed: some of the mentioned theorems

were proved in this way by S . P. Lalley in [La].

Entropy. Another logical thing to expect is that this measure will

maximize the entropy. Indeed, this was proved by M. Lyubich (see

[Lyl-31): particularly, there exists a unique invariant measure maxi-

mizing the entropy, it coincides with balanced measure and has entropy

log deg F, which is the topological entropy of this dynamical system.

The latter equality was conjectured earlier by R. Bowen in [Boll, and

partial results in establishing it were obtained by M. Jakobson, J. Guck-

enheimer, M. Misiurewicz, and F. Przytycki (see [Jl-31, [Gu], [MP]).

Moreover, this balanced measure has very nice ergodic properties:

in the case of totally disconnected Julia set it is Gibbs (see e.g. [C] and

[MV] for more general setting of conformal Cantor sets), for hyperbolic

polynomials the mixing is exponentially fast.

1.2. Geometric Measures

One knows that self-similar sets (e.g. Cantor sets, snowflakes, etc.

- see [Fa]) usually carry some geometric measure - like Hausdorff or

Minkowski - since they have nice scaling properties. It is logical to

expect the same from the Julia set.

Since the Julia set has more complicated origins of self-similarity

than, say, an affine cantor set, it is hard to evaluate (even estimate)

its Hausdorff dimension and understand geometrically what will be the

proper Hausdorff measure. One has to approach this question dynam-

ically, as was first done by D. Sullivan. If we assume that the Julia set

has Hausdorff dimension t and, moreover 0 < fit ( J F ) < oo, then the

Jacobian of the Hausdorff measure fit (or normalized v : = fit / X t ( J F ) )

will be equal to IF'^^:

1 F' l t du = u ( F ( A ) ) , if F is injective on A .

Therefore one can look for a probability measure on JF with such

property, and hope that it will be a t-dimensional Hausdorff or some

other geometric measure. We will call such measures t-conformal.

Patterson-Sullivan construction. A constructive method to find

conformal measure, comes from the theory of Kleinian groups, where it

was introduced by S. J. Patterson [Pal] and then used by D. Sullivan

[Su1,3] (see also exposition of related results in [Pa2]). It is pretty

much the same as the described method for constructing a balanced

measure: we fix t , pick some point z outside the Julia set, and put a

unit mass on it. To "make" this measure conformal, we have to put

appropriate masses on preimages of z , then on their preimages, etc.

and consider a weak limit. Roughly speaking, we take

and set v to be the weak limit of the normalized measures vn /Var (vn) .

In practice the process is more, difficult: one has to chose t depending on

the convergence of the sum above, and sometimes "correct" the terms

in it to make the weak limit conformal. D. Sullivan has shown that for

any rational map there exists t for which we can construct a conformal

measure in such a way.

Particularly, there exists a 6-conformal measure for

for some (any) 2 outside J F . The latter series is an analogue of the

Poincar6 series for a Kleinian group.

Geometr ic propert ies of conformal measures. Unfortunately it

is difficult to deduce much about the conformal measure from the con-

struction or conformal property itself in the general case. There might

not be unique S with existing conformal measure, hypothetically con-

formal measure for given S might not be unique, it might be atomic

and does not have much of a geometric meaning.

First results were obtain by D. Sullivan, [ S U ~ ] , in hyperbolic case,

following the work [Bo2] of R. Bowen on the dimension of quasicircles

(see about Bowen's formula later). Since every small ball is mapped

eventually by some iterate of F to large scale with bounded distortion,

and after some work one obtains that

for any ball B, of radius r centered on the Julia set. Thereafter v is

(up to a constant) equal to a &dimensional Hausdorff measure, 6 being

the Hausdorff dimension of JF . In this case there is a unique conformal

measure for a unique exponent.

In the general case, results are harder and many things are still

unknown. If we set 6 to be a minimum exponent with an existing con-

formal measure, it is unknown whether 6 is always equal to Hausdorff,

Minkowski or hyperbolic dimension of JF. The latter dimension is de-

fined as a supremum of the Hausdorff dimensions of hyperbolic subsets

of JF, where dynamics is expanding.

However, some partial results are known. M. Denker and M. Urbari-

ski have shown in [DU5] that the following numbers coincide: 1) mini-

mal zero of the pressure function, 2) supremum of Hausdorff dimensions

of ergodic invariant measures with positive entropy (dynamical dimen-

sion), and 3) the minimal exponent 6' for which a measure conformal

except on some finite set exists, coincide. Moreover, for many rational

maps 6' and the minimal exponent 6 for conformal measures are the

same (some kind of "expanding" on critical orbits is sufficient).

If there are no recurrent critical points in the Julia set (but we allow

parabolic points), the situation is even nicer - M. Urbariski has shown

in [U] (partial results were obtained earlier by him and M. Denker

in [DU2-4,7]) that the exponent S will coincide with Hausdorff and

packing dimensions of the Julia set and the 6-conformal measure will be

equal to normalized &dimensional Hausdorff or /and packing measure.

The latter is determined by the existence of parabolic points, which can

produce interesting dichotomies for the possible properties of conformal

and invariant measures, see [ADU] .

It remains to mention that work [Sh] of M. Shishikura implies that

for topologically generic quadratics with parameter in the boundary of

Mandelbrot set all mentioned dimensions will be equal to 2. However,

we don't know what will be the 2-conformal measure in this case if the

Julia set has zero area.

1.3. Transfer Operator

It is plausible to have another approach, maybe not working for a

general Julia set, but giving more properties of a conformal measure.

First we introduce a notion of (A, t)-conformal measure, i.e. such mea-

sure v that

A F " i d v = v ( F ( A ) ) , if F is injective on A .

Note that it generalizes two previous definitions: t-conformal measure

is (1, t)-conformal, whereas balanced is (d, 0)-conformal.

This notion is very closely related to one of a transfer (Perron-

Frobenius-Ruelle) operator. Transfer operator with weight 4 is defined

in a proper functional space. Choice of it is very important and will be

discussed later.

The main role is played by the family Lt of transfer operators with

weights #t := I F'J-t . If the Julia set does not contain critical points,

Lt acts on the space of functions, continuous on JF. Hence its formal

adjoint acts on the space of Bore1 measures supported on JF:

It is easy to see that a measure v is (A, t)-conformal if and only if

it is an eigenmeasure of LZ; with eigenvalue A. Moreover, if Lt has

an eigenfunction f with the same eigenvalue, the measure f v is an

invariant measure equivalent to v.

Pressure. For hyperbolic polynomial let rF(t) denote the spectral

radius of the transfer operator Lt in a proper space (e.g. C( JF)). We

define the pressure by PF(t) = log rF(t). As we shall see, by carrying

out the spectral analysis of the transfer operator, one can establish

nice properties of pressure and connect it to other characteristics of the

Julia set.

For the Julia sets, pressure was first used by D. Ruelle in [Ru2] to

establish a conjecture of D. Sullivan that HDim JF depends real ana-

lytically on a hyperbolic rational function F. Particularly, he applied

the machinery of thermodynamic formalism (see [Ru1,4]) and showed

that when the Julia set JF of a rational function F is hyperbolic, the

pressure function PF(t) is real analytic as a function of t and F. The

remaining ingredient of his proof was

Bowen's formula. In the hyperbolic case the only zero of PF(t)

is the Hausdorff dimension HDim JF of the Julia set - the intuitive

reason is that conformal measure for t = HDim JF should coincide

with Hausdorff in the same dimension and hence At = 1 and PF (t) = 0.

This formula was noticed by R. Bowen for quasicircles, [Bo2] ; D. Ruelle

established it for hyperbolic Julia sets in [Ru2] and [Ru3] (see also

[DS] for another proof), it is equivalent to the mentioned result [Su2]

of D. Sullivan on conformal measures.

Spectral analysis of the transfer operator. In the hyperbolic case

the map F is expanding, so the transfer operator makes "smooth" func-

tions even "smoother" which implies its quasicompactness in the spaces

of smooth functions (e.g. Holder continuous - see later), and rF ( t ) is a

simple isolated eigenvalue with eigenfunction ft and eigenmeasure vt :

Operator Lt depends real analytically on t , thus by perturbation theory

PF (t) is real analytic as a function of t. Actually, in "infinitely smooth"

spaces (e.g. functions real analytic in the neighborhood of the Julia set)

the transfer operator behaves even nicer: D. Ruelle proved that it is

nuclear in the sense of A. Grot hendieck ( [Grl] ) , see [Ru2] .

Eigenfunction ft and eigenmeasure ut also depend real-analytically

(in a proper sense) on t and play an important role: vt is (r, t)-conformal

(e.g. for t = 0 it is balanced and for t = HDim JF is equal to the nor-

malized t-dimensional Hausdorff measure), ft vt gives us an equivalent

invariant measure.

Quasicompact ness. Another met hod of making spectral analysis

and establishing quasicompactness is due to C. T. Ionescu-Tulcea and

G. Marinescu (see [IM] and also the paper [N] of R. Nussbaum): the

"smoothing" property of the transfer operator leads us to the inequality

of the type

I I L n f IIsrnooth space 5 - &In I I f IIsrnooth space + An I I f IIusual space

which "pushes" the essential spectral radius in the "smooth space"

down to (A - E ) and makes operator L quasicompact.

By quasicompactness we mean that the essential spectral radius

re,, (L) is strictly less than the spectral radius r(L) of the transfer op-

erator L. Therefore the spectrum outside of the disk of radius re,,(L)

consists of finite number of isolated eigenvalues. We are interested in

the main eigenvalue r (L) , determining the pressure; under nice circum-

stances it appears to be an isolated eigenvalue of multiplicity 1, which

provides good properties of pressure.

Zeta-function. To prove the real analyticity of PF (t) in F, D. Ruelle

considered another approach, which gives the same pressure: he defined

PF (t) as the inverse to the pole of the dynamical 5-function

with the smallest modulus. Real analyticity follows then from nice spec-

tral properties of the transfer operator and general theory of Fredholm

determinants (see [Gr2]), to which C-function is closely connected.

Variational principle. Variational approach works in the general

setting too. For $ E C(JF) , the pressure is defined as

where the supremum is taken over all probability measures p on JF

invariant under F. In the hyperbolic case, this variational problem has

a unique solution for smooth $ and the pressure function is PF(t) =

P (-t log IF' I). Moreover, the maximizing measure for $ = -t log IF' 1 is equal to ftvt.

On computer experiments. Nice spectral properties of the transfer

operator imply that, iterating it, we converge to the main eigenfunction.

Particularly, in the hyperbolic case, the partition function 2, satisfies

giving us an opportunity to estimate the main eigenvalue numerically

(with an error of const '/" - 1 x 1 In, if we compute Ln by taking the

preimages of z under F-n).

Hence whenever the transfer operator behaves "nicely," particularly

in the hyperbolic case, one can make rigorous computer estimates of

all the parameters involved: spectra, dimensions, etc. - see [STV] and

[V] by G. Servizi, G. Turchetti, S. Vaienti. Moreover, reasonable algo-

rithms should converge exponentially fast, depending on the "expan-

sion" constants for the dynamics. See, e.g., paper [GI of L. Garnett for

computations of Hausdorff dimension for hyperbolic quadratics z2 + c

with small c. Roughly speaking, she evaluated zero of the pressure via

considering the finite rank approximations to the transfer operator and

computing their spectral radii.

Unfortunately, for most interesting cases with low hyperbolicity, the

exponent is close to 1 and can have a nasty constant in front of it,

spoiling the situation. So, with the present abilities of computers, it

still seems favorable to use "unrigorous" methods (e.g. Monte-Carlo),

speeding up the computations: consult the paper [BZ] of 0. Bodart

and M. Zinsmeister.

1.4. Non-hyperbolic situation

The key to analysis of spectra and properties of JF lies in the spectral

analysis of the transfer operator. If the dynamics is "expanding" in

some sense, the transfer operator has a nice spectrum (in a proper

space), its eigenvalues and eigenfunctions (eigenmeasures of its adjoint)

behave "nicely" and all mentioned objects are "nicely" connected and

have "good" properties. The main problem is hence to introduce a

proper notion of "expanding," and analyze the spectrum of transfer

operator in a proper space - the rest will follow.

In a non-hyperbolic situation there are many difficulties, since the

dynamics is not expanding in the usual sense. Moreover, some of the

previously mentioned definitions must be modified in this case (e.g. if

there are critical points on the Julia set, the transfer operator does not

act on C ( J F ) for t > 0) and a priori they can lead to different objects.

Since the first work of Ruelle, some progress has been made for

the case of expansive maps, which corresponds to the Julia sets with

parabolic points, see e.g. [HR], [Ru~] , [ADU] or [DU2,3,6,7].

In the case of an arbitrary rational function M. Denker, M. Urbariski,

and F. Przytycki (see [Dull, [Prl], [DPU]) showed that transfer op-

erator L$ with a Holder continuous, positive 1C, is almost periodic on the

space of Holder continuous functions (under an additional assumption

(0) -see the Subsection 2.3). This is a weaker property than quasicom-

pactness, and if there are critical points on the Julia set, Lt = LIP, , - t

for t # 0 does not satisfy their assumptions. Recently N. Haydn has

proved a stronger theorem, establishing the quasicompactness of the

transfer operator in this case - see [Ha].

An interesting partial case consists of polynomials z2 - c with real

c < -2. Then the Julia set is a Cantor subset of the real line, and

the potential 1 F' ( z ) l M t , restricted to it, is equal to a holomorphic func-

tion ~ ' ( z ) - ~ . This was used by A. Eremenko, G. Levin, M. Sodin,

and P. Yuditski'i in [ELS] and [LSYl-21 to thoroughly investigate the

spectrum of the transfer operator with t = 2.

Substantial progress has been made on similar questions in one-

dimensional dynamics, mainly using bounded variation spaces, see,

e.g., [HKl-21, [Pol], [Ru~] , [Ry] (see book [RuS] of D. Ruelle for

exposition of this subject). There is no good analogue of the bounded

variation space for the complex plane; BV on the interval is invariant

under monotone transformations, while in the complex plane we can

only hope to find a space invariant under conformal maps, and still

we will have some problems with critical points. Nevertheless, consid-

ering the space BV2 of functions whose second partial derivatives are

measures, D. Ruelle [Ru6] proved the quasicompactness of L$ with

II, E BV2 assuming certain conditions. His theorem implies some par-

tial results for t he negative spectrum.

Another way is in using Markov extensions, introduced by F. Hof-

bauer in [Ho] and developed by G. Keller in [K]. This technique is very

much related to the "jump transformation," when instead of given dy-

namics for every point one considers a proper iterate, carrying enough

expansion (such method usually works in expansive case - see, e.g.

[ADU]). The idea is to do the same thing, but "separating" different

iterates - we make a "tower" of countably many copies of the original

dynamical system: the point goes up, till it accumulates enough expan-

sion, then returns to the first floor. Hofbauer towers were used very

successfully in analyzing maps of the interval, see papers [BK], [KN],

[HK3-41, and thesis [Bru] of H. Bruin, which contains very instructive

exposition of the tower construction. We will use Markov extensions

build on the Yoccoz puzzle to analyze the non-recurrent quadratics.

Relating the pressure to the C-function is more difficult (see, e.g.,

[Ru5-6]), consult [Ba] and [Ru8] for the account of results in one-

dimensional dynamics. For non-hyperbolic Julia sets, we are aware only

of the work [Hi21 of A. Hinkkanen, who deduced an explicit formula

for the C-function with constant weights (i.e. t = 0), which appeared

to be always rational.

lb . Fine structure of measures There are a few other approaches of analytical and geometrical na-

ture, rather than dynamical, which lead to the same objects.

1.5. Complex analysis

Harmonic measure. One of the most important notions in the study

of the subsets of complex plane (more precisely, boundaries of domains)

is harmonic measure. For a given domain (and a point inside) one can

define harmonic measure as an equilibrium distribution for logarithmic

potential, probability of being hit by Brownian motion, or as an image

of the length on the unit circle under the Riemann uniformization map

for simply connected domains. There are other definitions, generating

more applications of this notion; we refer the reader to the monograph

[GM] of J. Garnett and D. Marshall. The choice of the point is not

important for the local behavior, so we will consider the harmonic mea-

sure on the boundary of the basin of attraction to infinity with respect

to infinity, which we will denote by w.

In the past decade there was a lot of progress in understanding geo-

metric behavior and fine structure of harmonic measure, see exposition

and references in the lectures [Mak2] of N. Makarov. Of particular

interest to us are the papers [CJ], [JM], and [CM] of L. Carleson,

P. Jones, and N. Makarov, where they obtained bounds on the ex-

tremal behavior of harmonic measure. These papers show that it can

be approximated on self-similar sets, which have good scaling proper-

ties and hence nice fine structure of harmonic measure. One should

note, that many of its properties were first (and easier) observed and

proved for self-similar sets: e.g. the dimension of harmonic measure on

the boundary of any simply connected domain by Makarov's theorem is

equal to 1 ([Makl]); the proof for domains of attraction to infinity (for

connected hyperbolic polynomial Julia sets) is easier and has more in-

tuitive reasons, than in general case - see paper [Man] of A. Manning.

This provides an additional motivation to investigate harmonic mea-

sure on self-similar fractals, since still there are many open questions

about its behavior in general case.

Distort ion of the Riemann map. Having a simply connected do-

main, one can learn about the geometry of the boundary by measuring

the distortion of the Riemann uniformization map. One possible way

is to consider the rate of growth of the integral means of its deriva-

tive, this makes even more sense if domain has self-similar boundary.

Particularly, for a connected polynomial Julia set we can define the

conformal spectrum in the following way:

log hzIZr l9'lt PF(t) := limsup 1

+ log, , t E R ,

where cp is the Riemann map from the outside of the unit disk to the

basin of attraction to oo. Considering the conjugation of F outside

JF with z zd outside the unit disk, given by properly normalized

cp, it is easy to see that in hyperbolic case PF(t) = (PF(t) - t + 1)

log (deg F).

There are also ways to extend this definition to the multiply con-

nected case, e.g. considering mean values of distance to the boundary

(or modulus of gradient of the Green's function), taken to some power,

along the Green's lines.

Harmonic Measure on Jul ia Sets. Many geometric properties of

Julia sets are closely related to the behavior of harmonic measure and

different spectra under discussion:

The dichotomy of A. Zdunik (see [Zd], for hyperbolic Julia sets it

was proved by F. Przytycki, M. Urbariski, and A. Zdunik in [PUZ]) for

the boundary of attractive basin to be either an analytic curve or have

Hausdorff dimension strictly more than 1, is proved using LIL (Law of

Iterated Logarithm) principle for harmonic measure. Her theorem is

in fact stronger, and has the following meaning in terms of conformal

@-spectrum: either the boundary is an analytic curve (then P r 0), or

the second derivative P" (0) is strictly positive.

The Pommerenke-Yoccoz-Levin-Petersen inequality, which estimates

the multipliers of periodic points in connected Julia sets (see [Porn],

[ELel-21, [Le] , [Pel), is related to the A. Beurling's estimate for the

possible concentration of harmonic measure at a point when we know

how "twisted" is our domain.

By [GS] and [CJY] we know that Collet-Eckmann condition (i.e.

exponential expansion on critical orbits) forces Fatou components to

be Holder, particularly semihyperbolic polynomials have Fatou compo-

nents even John domains. Holder property is closely related to spectra

and their regularity: simply connected domain is Holder if and only

if for the conformal spectrum limt-t+oo ,B(t)/t < 1 (equivalently if

and only if PF(t) < 0 for large t), the value of the limit depending on

the Holder exponent. In this case, by [Mak2], the only root of the

equation ,B(t) = t - 1 is the Minkowski dimension of the boundary (an

analogue of the R. Bowen's formula in non-dynamical context).

We want also to note that many things known about the geometric

regularity of the Julia sets imply some nice properties of harmonic

measure. R. Mafib and L. F. da Rocha have shown in [MR] (see also

[Hil l) that Julia sets of rational functions are uniformly perfect, this

property ensures some kind of regularity usually needed for the study

of harmonic measure on disconnected sets.

O n rat ional functions and conformal Cantor sets. We will

mainly discuss the harmonic measure on the basin of attraction to

infinity for polynomial Julia sets, which coincides with the balanced

measure of maximal entropy. In the general case one has to consider,

instead of a pressure as function of t , P = P ( t log IF'I), a pressure of

two parameters: P = P ( t log I F'I + s log J,), taking into account the

non-constant Jacobian J, of the harmonic measure.

However, many of our constructions do apply to conformal Can-

tor sets, or (super)attractive/parabolic basins of attraction for rational

functions. It appears that the situation is still good enough to estab-

lish the regularity of the fine structure, though some other things, like

rigidity properties, may "go wrong" (see e.g. [C], [MV], [LVl-21 for

questions concerning conformal Cantor sets).

1.6. Multifractal Analysis - Chaotic Sets

Multifractal analysis is an intensively developing subject on the bor-

der between mathematics and physics. It was introduced by T. Halsey,

M. Jensen, L. Kadanoff, I. Procaccia, and B. Shraiman in a physical

paper [HJKPS], where they tried to understand and describe scal-

ing laws of physical measures on different fractals of physical nature

(strange attractors, stochastic fractals like DLA, etc.) . They considered

the dimension spectrum of those measures - a continuum of parameters

characterizing the size of the set where certain power law applies to the

mass concentration.

Since then there appeared a number of papers with rigorous mul-

tifractal analysis of different dynamically defined measures, see e.g.

[PW] for references. For a physical approach to multifractal analysis

and its mathematical counterpart, consult [BS] , [Fa], [Fe] , [Mand] ,

and [T2]. It is natural to perform (and expect to perform well) mul-

tifractal analysis for self-similar sets and for measures behaving nicely

under rescaling. Generally it is done for expanding dynamical systems

(- hyperbolic Julia sets) and for Gibbs measures.

It appears, that the fine structure of harmonic measure is best de-

scribed via its multifractal analysis, see [Mak2]. A. Lopes LO^])

made mult ifract a1 analysis of equilibrium (harmonic) measure for hy-

perbolic Julia sets, when all reasonable definitions of spectra work and

lead to the same objects. He also pointed out in [Lo5-61 some ir-

regularities (to be discussed later), which may occur in subhyperbolic

situation. Also see papers of Y. Pesin and H. Weiss [PW] , [PI, who,

in particular, analyze equilibrium measures with Holder potentials for

conformal expanding case.

Dimension spectra. Dimension spectra will estimate the size of the

set where our measure has certain power law behavior. It makes sense

to consider two kinds of dimension spectra, of "Minkowski" and "Haus-

dorff" types. Roughly speaking, we define (for rigorous definitions see

[Mak2]) Hausdorff dimension spectrum as

J(a) := HDim { z : wB,,~ % S", 6-4) .

To define box dimension spectrum, we not just change the Hausdorff

dimension to box-counting, but also "flip" limits in the definitions of

spectrum and dimension:

f (a) := lim BDima { z : w B ~ , ~ x Sa) . 6-0

Here BDima is the box-counting dimension, estimated with disks of

radii 6:

BDims E := inf { p : 3 disjoint {B,,a), z E E with E S P =: 1) . B

Of course, in the general situation there will be many points, where

measure behaves differently at different scales, so we will have to add

lim sup's and lim inf's to our definitions.

By analyzing the multifractal structure of some measure we mean

investigating relationship of dimension spectra with other objects and

proving some kind of its regularity - like the possibility of taking real

limits (instead of upper or lower) in the definitions.

Packing and covering spectra. Sometimes it is more convenient to

play with disks, having certain concentration of harmonic measure, in a

different fashion, analogous to considering grand ensemble in statistical

mechanics (cf. [Rul] ) . Resulting spectra appear (under nice circum-

stances) to be connected via Legendre transform to dimension spectra

and behave more like pressure in dynamical situations. In fact, pres-

sure evaluates the same quantities with Lyapunov exponent s instead

of rescaling coefficients for measures; in dynamical context those are

closely related, as was noticed by J.-P. Eckmann and I. Procaccia in

[EP] (see also [FPT], [ST], and [TI]).

We define the packing spectrum ~ ( t ) as

q : VS> 0 3 6-packing {B) with ) : S ( B ) ~ ~ ( B ) ~ > 1

where S(B) is the diameter of disk B. Covering spectrum c( t ) is defined

similarly, as

q : V6 > 0 3 S - cover {B) with ~ ( B ) ~ U ( B ) ~ 5 1

In nice situations (e.g. hyperbolic Julia sets, there, in fact, also f - j and .rr - c) , spectra and dimension spectra are related via Legendre-

type transforms (see Figure 1) :

n(t) = sup f ( a ) - t , f (a ) = inf ( t + ax ( t ) ) , a>O a t

c ( t ) = sup ?(a) - t 7 f (a) = inf (t + ac(t)) .

a>O a t

For the properties in general case, see [Mak2].

h, = logd f ( a ) / a

A

f (4

HD JF -

/

0 C

a -t log d / P(t)

Figure 1. Spectra of hyperbolic polynomials

2. Results

Summing it up, one observes that spectra for negative values of

t (or small a ) describe the geometry of the set at "more exposed"

points - tips - with higher concentration of harmonic measure. The

more negative is t , the higher the concentration. On the other hand,

spectra for positive t (big a) describe the geometry of "fjords," where

the harmonic measure is low, but the set is "more dense" (here the

Hausdorff dimension "lives" ) .

A physical interpretation of this is discussed by T. Bohr, P. Cvi-

tanoviC, and M. H. Jensen in [BCJ], where they suggest that spectra

for small a should be robust under small parameter perturbation, when

large a ' s are noisy and poorly convergent - the fjords are screened, and

this can manifest in a "phase transition" at the Hausdorff dimension.

We prove nice behavior of the spectra for negative values of t for

any polynomial Julia set. All of them coincide, nicely converge and are

real analytic except for the rare class of polynomials with very specific

combinatorics of the critical orbits, which causes their Julia sets to have

points with unique geometry, unrepeated elsewhere. In some cases this

phenomenon can cause the "Hausdorff" and "Minkowski" spectra to be

different, with latter having a "phase transition." The reason is that

"Hausdorff" definitions neglect the input from individual, not repeated

patterns, while "Minkowski" take them into account.

For positive spectra the situation is indeed more difficult - lack of

expansion has more impact: when the critical point belongs to the Julia

set the weight JF '(z)~-~ is not even bounded, and we have troubles

defining the pressure. However, using different methods, we were able

to work out three particular cases.

For parabolic Julia sets (dynamics is not expanding, but is expansive,

and there are no critical points on the Julia set) we prove nice behavior

of spectra up to the value t = HDim JF, where the phase transition

happens. In this case it is caused not by a unique pattern, but by

infinitely duplicated cusp at the parabolic point. Then work [ADU]

of J. Aaronson, M. Denker, and M. Urbaliski implies an intriguing

dichotomy for discontinuity of the derivative at the phase transition

point and the value of HDim JF.

For subhyperbolic Julia sets (M. Misiurewicz - W. Thurston sets

where there can be preperiodic critical points and the dynamics is ex-

panding with respect to a metric with finite number of singularities)

we were able to transfer the problem via the Riemann uniformization

map to the unit circle, where preperiodicity of the critical point allows

to multiply the weight function by a proper homology, thus getting rid

of its singularities. Resulting pressure behaves nicely and has proper

connections to other spectra.

In a more difficult semihyperbolic case (M. Misiurewicz's Julia sets

where dynamics is expanding with respect to some singular metric -

see paper [CJY] of L. Carleson, P. Jones, and J.-C. Yoccoz for other

equivalent definitions and proof of the John property for their Fatou

components) we constructed, for non-recurrent quadratics, Markov ex-

tension (analogous to F. Hofbauer tower) on the J.-C. Yoccoz puzzle

(see J . Hubbard's [Hu] and J . Milnor's [Mi21 expositions of the J.-

C. Yoccoz's results). Dynamics on this tower is expanding, hence spec-

tra for tower behave nicely and have good relation to some spectra for

the original dynamical system.

2.1. Negative spectra

Establishing the quasicompact ness of the transfer operat or on Sob-

olev spaces (see Chapter 2 for precise formulation), we prove the fol-

lowing results, which almost completes the analysis of the negative

spectra.

Theorem A. A l . For any polynomial F and negative t either

(i) PF(t) is real analytic on (-oo,O), or

(ii) there exists a "hase transition'' point to < 0 such that

PF ( t ) is real analytic on [to, 0) , and

= - P-, t on (-oo,to] .

In fact PF (t) = max(- P-, . t , PF ( t ) ) for some FF ( t ) real an-

alytic on (-00, 0).

A2. In the case (i) all mentioned definitions of pressure give the same

function PF( t ) . In the case (ii) function PF( t ) corresponds to "Box"

definitions, when "hidden" spectrum &(t) - to "HausdorEn All defi-

nitions converge nicely, i.e. one can take lim instead of limsup, etc.

A3. For the occurrence of a "hase transition" i t is necessary that

either

(iia) F is conjugate to a Chebyshevpolynomial (then JF is an inter-

val), or

(iib) there are a fixed point a, F a = a, and a positive number E such

that

where p(b) : = I (F")' ( b ) I ' denotes the multiplier of a periodic

point b. This implies that a E J F , P-, = logp(a) and

~ - ' a \ { a ) c Crit F ( = zeroesof F ' ) .

A4. For quadratic polynomials (ii b) cannot happen (for com binatorial

reasons), and for cubics it only happens for some polynomials with

disconnected Julia set. However, there is a degree 4 su bhyperbolic

polynomial with connected Julia set for which i t occurs.

A5. The pressure function depends continuously on F as a function in

the space Cm(-00, 0)) or Coo ((-00, to - E) U (to + E, 0)) if the phase

transition occurs.

Parts A1 and A2 state that the distribution of the harmonic measure

at the "more exposed" points is very regular, and multifractal analysis

does apply. Part A3 and A4 show that the phase transition is a rather

rare phenomenon. The Julia set JF itself does not depend continuously

on F , nevertheless A5 shows' that "more exposed" parts of the Julia

set are "rigid" and depend continuously on the dynamics.

The geometrical meaning of the phase transition is that the tip at

point a is unique and "significantly more exposed" than any other point

of the Julia set. Thus for t > to the spectrum is determined by the

combined influence of many similar parts, but for t 2 the input from the

"very exposed" tip at point "a" "screens" the rest of the real-analytic

spectrum (which, nevertheless, continues to exist and can be calculated

as Hausdorff spectrum).

The "thermodynamical" meaning of the phase transition is that for

each t > to we have a unique equilibrium state, supported on the whole

Julia set. For t = to we obtain another equilibrium state, given by

a 6-measure at the point a, which dominates for t < to. However,

the original equilibrium state "continues" to exist being hidden. For

Chebyshev quadratic (F(z) = z2 - 2 and Julia set is an interval) this

phenomenon was observed by E. Ott, W. Withers,and J. A. Yorke in

[ O W ] and A. Lopes in [Lo5,6]. Also in the latter paper maps having

the so called gap (see [Lo51 and compare to the maps falling into the

case (ii) of Theorem A) are discussed, including Chebyshev polynomial

and the Lattes example (F(z) = ((2 - 2 ) / ~ ) ~ , Julia set is the whole

sphere).

Figure 2. Phase transition in negative spectra

In other words, the transfer operator is always quasicompact, and

the maximal eigenvalue is isolated, but for to two eigenvalues (one gen-

erated by a nice measure, and another - by a 6-measure at the tip)

"cross" and the maximal eigenvalue has multiplicity 2. Note that this

differs from the phase transition in positive spectrum (e.g. for parabol-

ics), or one described by M. Feigenbaum, I. Procaccia, and T. T61 in

[FPT], when the essential spectral radius reaches the maximal eigen-

value. In our case the transfer operator stays quasicompact and hence

spectra behave nicely.

2.2. Positive spectrum

Parabolic case. Assume that all critical points are attracted para-

bolic or (super) attractive cycles. Establishing the quasicompactness

of transfer operator for the parameter values t < HDim JF, we prove

the following

Theorem B. B1. For a parabolic polynomial F, the function PF (t)

is real analytic on [0, HDim JF) and PF (t) = 0, t E [HDim JF, +oo).

B2. The derivative of PF (t) is discontinuous a t the point HDim JF if

and only if

L

HDim JF > 2 - - p + l '

where p is the maximal number of petals a t the parabolic points.

The reason for the phase transition is that, while for t < 6 transfer

operator is quasicompact and conformal measure is "nice", for t > 6

the situation changes, conformal measure becomes atomic - supported

on the preimages of the parabolic cycle. Another interpretation is that

main input in the spectra for t > HDim JF comes from the infinitely

many times duplicated cusp at the parabolic point.

The dichotomy comes from the work [ADU] of J. Aaronson, M. Den-

ker, and M. Urbaliski on the existence of the invariant measures equiv-

alent to 6-conformal.

Figure 3. Spectra of parabolics with HDJf > 2 - &

2 Figure 4. Spectra of parabolics with HD J f 5 2 - 3

The inequality (4) is true for z2 + 114 (the "cauliflower" Julia set -

indeed, its dimension is greater than I), and hence there is a disconti-

nuity of the derivative at the phase transition point. We do not know

which case of the dichotomy occurs even for other parabolic quadratics;

and we were not able to prove the discontinuity of the derivative for

z2 + 1 /4 directly. Note that computer experiments in [BZ] suggest that

for z2 - 314 the inequality (4) fails.

Note that as an application of the quasicompactness result we obtain

an improved estimate on the radius of essential spectrum of the transfer

operator (in hyperbolic or parabolic case), which depends only on t,

P( t ) , and P(f oo) (see the Remark 1.7 in the corresponding Chapter).

Subhyperbolic case. Assume that all critical points in the Julia set

are strictly preperiodic. Transferring the problem to the unit circle, we

prove for the critically finite polynomials the following

Theorem C. For a su bhyperbolic polynomial F with connected Julia

set, the function PF (t) is real analytic on [O, +m) .

Semihyperbolic case. For semihyperbolic quadratic polynomials (i.e.

those with non-recurrent critical point) we build a Markov extension

of the original system (analogous to the Hofbauer tower), taking the

Yoccoz puzzle as a base. Then we establish the quasicompactness of

the corresponding transfer operator, and prove the following

CHAPTER 1. INTRODUCTION 39

2 Theorem D. For a non-recurrent quadratic polynomial F(z) = z + c

either

(i) PF (t) is real analytic on [0, +m), or

(ii) A phase transition occurs: there exists to > HDim JF such that

PF(t) is real analytic on [O,to), and

PF(t) = - l t 2 log (liminf,,, (F~) ' (C) ' ' " ) on [to, +m) .

We do not know whether case (ii) can occur. By the work [CJY] of

L. Carleson, P. Jones and J.-C. Yoccoz the polynomials under consid-

eration have domain of attraction to infinity A(m) John, thus one can

speculate that spectrum should be "nice" and (ii) impossible. On the

other hand, by [GS], Collet-Eckmann polynomials have Holder A(m),

so P(t) < 0 for large t and one can expect to have (ii) for Collet-

Eckmann recurrent polynomials.

2.3. Transfer operator on Sobolev spaces

We will apply the methods used for the analysis of negative spec-

trum to investigating transfer operators with general Sobolev weights.

Considering functional spaces on the complex sphere, or some open set

fl with

taken with spherical metric, we define transfer operator on the space

of continuous functions by

Lf ( z ) := C f ( Y ) g ( y )

where preimages y of z are counted with multiplicities. Note that

Lnf (z ) := C f ( ~ ) g n ( ~ ) 7

yEF-*y

where g,(y) = g ( y ) g ( F y ) . . . g ( ~ n - l ( y ) . Denote by X the spectral

radius:

We prove the following theorem, establishing the quasicompactness

of L:

Theorem E. Suppose that

for some (any) large n. Then for p > 2 sufficiently close to 2 operator

L acts on W 1 , and

ress (L) Wl,,) < r (L, W1,,) = X ,

where r and re,, are spectral and essential spectral radii of L as an

operator on Wl,,. Moreover, X is an isolated eigenvalue of multiplicity

one.

This theorem is analogous to the results [Dull , [Prl], [DPU] of

M. Denker, M. Urbariski, and F. Przytycki for the transfer operator

on the space of Holder continuous functions They have established the

(weaker) property of almost periodicity under the assumption (0). Re-

cently N. Haydn has extended their results, proving the quasicompact-

ness (see [Ha]).

2.4. Open problems

Negative spectrum. Some work still remains to be done for negative

values of t . If we establish a connection between the spectrum and

the C-function (the similar problem is stated in [Ru~]) , we might be

able to learn more about the dependence of PF(t) on F and get some

necessary and sufficient conditions for the phase transition to occur.

Particularly it is interesting whether or not (iib) is sufficient for the

phase transition (it is so in degrees 2 and 3).

Positive spectrum. A very optimistic statement is

Conjecture. For any polynomial F either

(I) PF(t) is real analytic on [0, +m), or

(11) PF (t) is real analytic on [O, tii) and PF (t) = P+t, t E [to, +w),

or

(111) PF ( t ) is real analytic on [O, tiii) and PF(t) = 0, t E [to, +oo).

A possible step in this direction is to establish

Conjecture. For any polynomial F, PF(t) is real analytic in some

neighborhood of 0.

Work [Zd] of A. Zdunik implies that for any polynomial (except

ones with Julia set being an interval or a circle) the pressure at zero has

strictly positive second derivative, which advances us in that direction.

Next step will be to find some necessary and (or) sufficient conditions

for (I), (11), (111) to hold: in terms of the orbits of the critical points

of F or geometry of the Julia set J F . There are some euristic reasons

to expect that case of John domain of attraction to infinity (i.e. semi-

hyperbolic polynomials - see [CJY]) corresponds to (I), while Holder

(but not John, e.g. Collet-Eckmann polynomials fall into this category,

see [GS]) case - to (11).

If the phase transition occurs, we need to analyze its nature. One ex-

ample is the question about the meaning of the first zero of the pressure

function (the "phase transition point" in case (111)). It is interesting to

compare it with such parameters as the dynamical, Hausdorff and hy-

perbolic dimensions of J F . There is some hope that in cases (I), (11)

all of these dimensions will coincide. On the other hand, the geometry

of the Julia sets with the phase transition (case (111)) is likely to be

"bad" and it might happen that the dimensions above will differ.

Dependence on F. The logical statement to prove is that PF(t)

depends continuously on F whenever it is positive. As in the case of the

negative spectrum, it might follow from the "sufficiently good" spectral

analysis of the transfer operator. If one wants to prove some stronger

results, like analyticity, he will probably need to study (-function.

2.5. About methods and organization

The general scheme of the proofs is to find a proper notion of ex-

panding, establish the quasicompactness of the transfer operator, and

deduce the results about the spectra. The thesis is organized as three

(independent) chapters, devoted to negative spectrum, parabolic and

semihyperbolic polynomials. Some arguments are similar in different

cases, but st ill they have (sometimes substantial) differences, so we

repeat them for the sake of completeness.

The Chapter 2 is devoted to the analysis of negative spectra for

arbitrary polynomials (Theorem A). In the arbitrary case F is not

expanding and Lt does not act on the space of Holder continuous func-

tions. This problem for t < 0 is solved by considering the Sobolev

space, where bad behavior of F near critical points is compensated by

/ -t the zeroes of the weight function IF I . Then we apply the same tech-

nique to the study of transfer operators with arbitrary Sobolev weights,

establishing the Theorem E.

In the Chapter 3 we consider the parabolic case (Theorem B), when

dynamics is not expanding but expansive, and careful computation

shows that non-expanding branches of F-" do not contribute much to

the transfer operator for t < HDim J F , hence spectra are good for the

corresponding parameters.

For other Julia sets and t > 0 the situation is more complicated,

since the function I F ' I - ~ is unbounded and thus Lt does not preserve

the space of bounded functions.

In the subhyperbolic case we "cheat" by multiplying the weight by

a precisely defined homology and cancelling its singularities. Then the

problem is transferred via the Riemann uniformization map to the unit

circle, where the corresponding dynamics is expanding. The pressure

for the new operator gives the same spectra. The corresponding result

(Theorem C) is proved in the paper [MS] by N. Makarov and the

author. We will also rely heavily on this paper during our analysis of

the phase transition phenomenon in the Chapter 2 (negative spectrum).

In the Chapter 4 we work (establishing Theorem D) with non-recur-

rent quadratics, building a Markov extension of original system - an

analogue of a Hofbauer tower in one-dimensional dynamics, which gives

the desired expansion.

CHAPTER 2. NEGATIVE SPECTRA

Chapter 2

Negative spectra

In this chapter we will consider the spectra for negative values of

parameter t , in which case transfer operator preserves the space of con-

tinuous functions. In the first Section we prove Ionescu-Tulcea and

Marinescu inequality and establish the quasicompactness of transfer

operator in the Sobolev space, while the second is devoted to the ana-

lyticity of spectra and their properties, which result in the following

Theorem A. A l . For any polynomial F and negative t either

(i) PF (t) is real analytic on (-00, 0) , or

(ii) there exists a phase transition" point to < 0 such that

PF(t) is real analytic on [tO,O) , and - - - P-, t on (-00, to] .

In fact PF (t) = max(- P-, t , pF ( t ) ) for some & (t) real an-

alytic on ( -00, 0) .

A2. In the case (i) all mentioned definitions of pressure give the same

function PF ( t ) . In the case (ii) function PF ( t ) corresponds to "Box"

definitions, when "hidden" spectrum PF (t) - to ('HausdorK " All defi-

nitions converge nicely, i.e. one can take lim instead of limsup, etc.

A3. For the occurrence of a (@phase transition" it is necessary that

either

(iia) F is conjugate to a Chebyshev polynomial (then JF is an inter-

val), or

(iib) there are a fixed point a, F a = a, and a positive number E such

that

1

where p(b) : = I ( F ~ ) ' ( b ) I denotes the multiplier of a periodic

point b. This implies that a E JF, P-, = logp(a) and

la \ {a) c Crit F ( = zeroes of F' ) . (6)

A4. For quadratic polynomials (iib) cannot happen (for combinatorial

reasons), and for cubics it only happens for some polynomials with

disconnected Julia set. However, there is a degree 4 subhyperbolic

polynomial with connected Julia set for which i t occurs.

A5. The pressure function depends continuously on F as a function in

the space Cw (-00, 0), or Cw ((-00, to - E ) U (to + E , 0)) if the phase

transition occurs.

Finally, in the last Section we apply the developed technique to the

study of transfer operators with arbitrary Sobolev weights. This results

in the Theorem E (see the Subsection 3.1 for more details).

1. Quasicompactness

1.1. Notation

Let S1 be a big disk compactly containing JF such that

and orbits of the critical points do not intersect 80.

For t < 0 we define operator Lt on C (a) by

where preimages y of z are counted with multiplicities. Note that

(here and later we write Fn = Fn ).

Denote

s ( t ) := logd r ( L ~ , C (n)) (spectral radius) .

It will be shown later that for connected JF

We will also consider this operator in various Sobolev spaces W1,,(0).

Observe that for p > 2, W1,, (0) c C (a) (see [Zi] for this and other

properties of Sobolev spaces).

1.2. Quasicompactness theorem

Theorem. For any negative t for p > 2 suficiently close to 2 operator

Lt acts on W1,,(O) and

r e s s (Lt , Wl,, (0)) < r (Lt WI,, (a)) = ds(t) ,

where r and re,, are spectral and essential spectral radii of Lt as an

operator on Wl ,, (0) .

1.3. Sobolev spaces

We consider Sobolev space W1,,(O) with the norm

If p > 2, then

Wl,P(fl) c C ( Q ) 9 and llfllm 5 Ilfll,

Moreover, Wl,p-functions are Holder continuous: Wl lp c Hall - 2 . P

More precisely, if 1 z - y 1 = 6, then

For the proofs of these results see W.Ziemer "Weakly Differentiable

Functions", particularly [Zi, 2.4.41.

Lemma. I f t < -2(1 - z ) , then LtWllp c W I , ~ .

Proof. It is sufficient to estimate

by const 1 1 f 1 1 l,p. Clearly

Here

< const / a f i P 7

since t < -2(1- :) implies - tp+2-p > 0 and supn IF'I --tp+a-p < 00.

The second term can be estimated by

so we just need to prove the convergence of the last integral which is

non-trivial only at the critical points of F.

Consider some critical point (say zero), let F' have singularity (zero)

of order k :

Then

Since k 2 1, t < -(1 - :)(I+ i ) thus

-p(l + kt) + k ( 2 - p) > -2 ,

which implies the convergence and hence the desired estimate.

1.4. Integral means ( t 5 0)

Lemma. For any (all) z E dS1

In JF is connected, then

CHAPTER 2. NEGATIVE SPECTRA

Remark. Lemma implies that P(t) is well-defined and

Proof. Clearly IILyllco = IILylllco. The function z I-+ L r l (z) is sub-

harmonic. Therefore

Fix zo E DR. For any z E dR we can choose a domain 9 3 z, zo

without forward iterates of critical points. Then any branch of FWn is

conformal on Q, so

where y, yo are images of z , zo under the same branch of F+. We can

estimate the distortion by a constant independent of z, hence

sup L; 1 x L,nl(zo) x L; 1 (2) , an

which implies the first statement of the Lemma.

Note that similar estimates (and F-n 0 \ KF = filled Julia set)

imply that

IF; (yo)l 2 1 for yo E FVnzo and large n ,

thus for t < u < 0

and

Moreover area estimate shows that

implying that for fixed d > q > 1 for large n most of the y E F - n ~ o

satisfy I FA ( y ) 1 > qn. Therefore for n >> 1

A - - -- n logq LYl(zo) . 2

So for 1

+n(t) := logd (Ly l (20 ) )

we have +,(t) < -$ log q < 0 which implies that s ( t ) is strictly de-

creasing because +, -+ s. n-oo

To prove the second part note that without loss of generality cp

d conjugates F with dynamics T : z H z on ID)-:

Differentiating the identity Fn o cp = cp o Tn, we obtain

Applying this equality to the preimages < E T-n of some fixed point

J E ID- we notice that the right side is =: dn, thus (taking to power t )

where y = p(C) is a corresponding preimage of z = p(c) under F n .

The points [ are equidistributed on the circle

with (rn - 1) =: - dn • Therefore summing over all C E TVn[ ( y E

F - n ~ ) , we have

which together with the first statement completes the proof.

1.5. Ionescu-Tulcea and Marinescu

inequality

Lemma. Suppose t < -2(1 - g), so Lt acts on W1,,(SZ). Then

CHAPTER 2. NEGATIVE SPECTRA

with

Proof. Clearly

We can estimate the right-hand side by

(in (*) we used H6lder inequality: (C aibi)' < C a: (C b f ) ' ) , and

I I P = A ( C f (Y) ~ Y I F ; ( Y ) I - ~ I F ; ( Y ) I - ~ d m 2 (2) y € F W n ( z )

Here Cn is finite since we assumed t < -2(1 - $) - use the same

estimate as in 1.3. It remains to notice that (see 1.4)

1.6. Finite rank approximation

Lemma. For any E > 0 there exists a finite rank operator M in

W1,&2) such that

IIMII1,, 5 abs. const ,

Proof. Cover 0 with a grid of triangles 4 of size S << 1. Define M f as

a continuous function satisfying

f at all vertices , M f = { is linear in each triangle 4 .

For a fixed 4, we have

and by the Holder continuity (see 1.3) this is

Taking to the power p and integrating over R (i.e. summing up over

all 4 's ) we have

which proves the first inequality.

The second one follows from the Holder continuity.

1.7. Analysis of w(p,t)

We show here that for any negative t < -2 1 - with p > 2 ( $1 function w (p, t ) satisfies

Clearly s(t) satisfies following two properties:

s(lct) 5 lc s(t) for lc > 1 and

s(t) is decreasing .

CHAPTER 2. NEGATIVE SPECTRA

.o(n>

Proof. In fact, d N S @ ) =: 1 1 L ~ ~ I I ml so obvious

implies the first property.

For the second see Proof of the Lemma 1.4. There we proved even

that s ( t ) is strictly decreasing but we use the simpler property for the

sake of 2.7.

Now we can write

If the second inequality is strict, we are done. Otherwise for any k E

[ I , P'I k

k s ( t ) = - s (p't) 5 s ( k t ) 5 k s ( t ) , P'

hence s ( k t ) = k s ( t ) and s ( t ) is strictly decreasing on ip't, t] . This

implies that the first inequality is strict and again we are done. ¤

1.8. Quasicompactness

Claim. Suppose t < -2(1- g) and p > 2. Then

r(Lt , Wl,,(R)) = dS@) ,

CHAPTER 2. NEGATIVE SPECTRA

and

Remark. Clearly p E (2, &) for t E (-2,O) and any p > 2 for t 5 -2

satisfies the condition of this Claim and therefore it completes the Proof

of our Theorem.

Proof. First recall that

Also

llL;llw llL;lllm 5 llL:llll,p 5 llL~lll,p 7

and ds(t) < r(Lt, Wl,p(R)).

To prove the converse, by the Ionescu-Tulcea and Marinescu inequal-

ity choose E > 0 and N such that

IILNf 5 dN(s(t)-') llf Ill,, + K llf ll,

Define inductively an increasing sequence { M k ) by

Then

CHAPTER 2. NEGATIVE SPECTRA

and we can choose a constant C satisfying

C dN(s ( t ) - " ) + 1 < - C d N S @ ) and K 5 C Ml .

By induction,

1 1 ~ ~ " f 5 dN"s(t)-') llf Ill,, + C M k llf ll, < dN"(t) N 1 1 f I l l,P

therefore

which implies the desired inequality and thus the first statement.

To prove the second statement choose q small such that

for any approximation operator ME. Since w(p, t ) < s ( t ) , we have

for some N and K. Take E < & d N S @ ) , and define M := ME. Then

CHAPTER 2. NEGATIVE SPECTRA

and

2. Analyticity

2.1. Remark on the proofs

We will prove Theorem A1 in the Subsections 2.3 - 2.8 below, last

two of which are also devoted to the analysis of the phase transition

phenomenon (Theorem A3). Multifractal analysis (Theorem A2) is

completed in the Subsections 2.2 and 2.9. In the remaining part of this

Subsection we prove Theorem A4-5.

Proof of Theorem Ad. Discussion in [MS, 6.21 shows that for combi-

natorial reasons (4) cannot happen for quadratic polynomials. Com-

parison of multipliers in [MS, 6.51 for cubic polynomials satisfying (4)

implies that iib) cannot be true for F with connected J F . See [MS, 6.61

for an example of a degree 4 subhyperbolic polynomial with a phase

transition.

Corollary. I f F is quadratic or cubic polynomial, it has a phase tran-

sition i f and only if (iia) or (iib) is true.

Proof of Theorem A5. Discussion below shows that the main eigenvalue

of the transfer operator is isolated and of multiplicity one for all t E

(-oo,O), or for t E ( -w, t0 - E ) U (to + E , O ) , if the phase transition

occurs. Using perturbation theory for transfer operators with weights

I F 'I-(~+'~), we conclude that for specified values of t pressure can be

extended to a holomorphic function of t + is in a thin strip -6 < s <

6 . By the perturbation theory again, this extended pressure depends

continuously on F, implying Theorem A5 via Cauchy formula. H

2.2. Variational principle

For t < 0 define pressure by

where supremum is taken over all probability measures p on JF invari-

ant under F. We also define P ( t ) as the same supremum, taken over

all probability non-atomic measures (or equivalently, with positive en-

tropy) *

Lemma. s(t) = P( t ) / log d and 5(t) = p( t ) / log d .

Proof. Equivalent property r (Lt ) = exp (P (t)) is mentioned in [Ru6,

6.31, where he refers to [Pr]. The same method works for the second

formula - see section 2.7 for the definition of S ( t ) .

2.3. Eigenvalues and multiplicity

Fix t and denote

Lemma. ker (Lt - A) # (0).

Proof. See [Ru6, Theorem 2.21.

Idea of the proof: since re,, (L t , W1,,(fl)) < r (Lt , W1,,(fl)) = A,

we have some eigenvalues Aj of absolute value A, and a decomposition

where Xo is the subspace corresponding to the rest of the spectrum.

Then

Here LTXo = o ( A n ) and LrX j = A? X j , implying that one of the

Xj7s is positive and hence is equal to X = r ( L ~ , C ( a ) ) .

Remark. This reasoning also implies that Lyl 5 An and therefore for

z E d o we have Lr 1 x An.

2.4. Conformal measures

Let L,* denote the adjoint of Lt : C ( J F ) 6.

Lemma. There exists probability measure v on JF such that

Proof. Fix a point z E afl and denote

Clearly L;pn = X pn+> Since

we have a following formula for p,:

and Var (p,) = A-" L?l(z) x 1.

Set

Then Var (v,) =: n and

Var (Lfvn - Xvn) = Var ( X / L , + ~ - Xpo) 5 1 .

Following Patterson and Sullivan choose a subsequence of normalized

measures vn / Var (vn) weakly converging to some measure v. Clearly

v is probability measure supported on J F . Since

Var (LTv, - XU,) / Var (u,) --+ 0 , n+oo

measure v satisfies Lfv = Xu.

Lemma. Suppose v is a measure from the previous Lemma. Then

suppv = JF unless F satisfies iia) or ( 4 ) .

Proof. Clearly v satisfies

hence z E suppv + F z E suppv and

F z E suppv , z 51 Crit F + z E suppv .

Then the same reasoning as in [MS, 6.11 proves this Lemma.

Main idea: preimages of any point z E JF are dense in J F , SO

if z E suppv does not belong to the forward orbits of critical points,

suppv = JF and we are done. The only cases when v can be supported

only on the forward orbits of critical points are iia) and (4).

2.5. Jordan cells

Example. Operator

acting on IK2 depends real analytically on t. But a (Qt ) = {f t ) and

therefore r ( Q t ) = It 1 is not a real analytic function.

Lemma. Suppose there exists an eigenmeasure v, L* v = X v with

suppv = J F . Then dim ker ( L t - x ) ~ = 1, i.e. X is a simple eigen-

value.

Proof. First we will prove the following

Sublemma. For f E Wl,p(52)

L t f = X f implies f = 0 .

Proof of Sublemma. For any z E 52 distortion estimates in 1.4 imply

~ ~ ; ( y ) l - l -- d-n lp ' (C) l x d-" dist (Y J F )

dist (c, aID-) x d-" dist ( Y , J F )

d-n = dist ( y , J F ) ,

where notation is taken from 1.4. Using Wl,p(fl) c Hol, with small

a < 1 - $ we obtain

because s ( t ) is strictly decreasing by 1.4. Consequently f = 0 and the

Sublemma is proved.

If we consider only functions in C (JF), the same reasoning as in

[MS, 3.61 gives us that dim ker (Lt - A) = 1, and then Sublemma

implies this for f E Wl,p (0).

Also [MS, 53.61 shows that we have an eigenfunction f non-negative

If A is not a simple eigenvalue, then there exists function h such that

Therefore

Thus v-a.e. we have f = 0, hence everywhere on JF and (by the

Sublemma) in R. ¤

2.6. Analyticity

Lemmas 1.8 and 2.5 show that if for some t' there exists eigenmea-

sure v, L;, v = Xt t v with suppv = JF, then for p > 2 close to 2

number ds@) is a simple isolated eigenvalue of Lt : W ( ) 6. Clearly

Lt depends real analytically on t therefore s(t) and hence P(t) is real

analytic in the neighborhood of t' (see [MS, 4.11).

So 2.5 implies that s(t) is real analytic on (-00, 0), unless iia) or (4)

happens.

In the case iia) JF is an interval, without loss of generality

and simple calculation shows that

It remains to deal with the case (4) when for some t' there is no

eigenmeasure supported on the whole J F . Then analysis in [MS, 61

implies that in fact vtt = 6, and At/ = p(a)-t (or equivalently s (t') =

-s-t' with s- = logdp(a) ). Set

By the proof of 1.4 s ( t ) > 1, thus to < 0.

For t E (to, 0 ) by the discussion above s ( t ) is real analytic on (to, 0).

Since

L,*S, = ,u(a)-t St = d-s-t 6, ,

we have s ( t ) > -s-t. On the other hand by 1.7 for t 5 to

t s ( t ) 5 - s ( to ) = s- t ,

t o

which implies that s ( t ) = -s-t for t E (-00, to].

Remembering the relation P ( t ) = s ( t ) + t - 1 we notice that to prove

the main Theorem it remains to show that s ( t ) can be extended from

[to, 0) to a real analytic function on ( -00 ,0) , and occurrence of the

phase transition together with (4) implies condition iib) .

2.7. Hidden spectrum

We want to analyze the case when F satisfies (4) and a phase tran-

sition occurs.

Let K := min {multiplicity c E F-'(a)). We will consider functions

G,,t := I F ' I - ~ Ha ,t H,,t F '

Ha,&) := Iz - alAat ,

and associated transfer operator

For all a! < 5 = * . amax function G,,t belongs to W1,JR) for p > 2

close to 2 and has zeroes of multiplicity 2 -tkmin with

at the critical points of F. Also denote

kmax := max {multiplicity c E Crit F) .

Set r,(t) := ( ~ a , t , C (a)) , s , ( t ) := logd r , ( t ) .

We can repeat all previous arguments for the operator L,,t with a! E

0 ) (or we can use results of the Section 3).

A few keypoints:

0 bviously

n-1

(G,,t), := G,,t o Fj = I F ' I - ~ Ha ,t , and j=o Ha,t 0 Fn

1 f ( 2 ) = jqq (L:,$H f) ( z ) ,

this relation helps to reformulate properties of L,,t in terms of Lt.

It means that considering L,,t we in fact restrict Lt to co-dimension

one subspace W1,, (0) fl { f : f (a) = 0) of W1,, (0) , "forgetting" the

eigenvalue of Lt which gives us the phase transition.

Then note that the same reasoning as in 1.4 shows that for 0 5 a 5

A and z E d 0 ~ + l

Since Ga,t has smaller orders of zeroes, to make La,t act on W1,,(fl)

we need a stronger condition, particularly

will be sufficient. But this inequality is equivalent to

which is clearly true for p > 2 close to 2.

An analog of w (p, t) < s(t) which we need for the proof of Ionescu-

Tulcea and Marinescu inequality is

< s,(t) with

This can be proved following 1.7, using the inequality sat ( t ) 5 s, ( t ) ,

which is true since

e o ( n >

T,l (t)n X LZI ,t 1 ( z )

= LZ,, Hal-, ( z )

for fixed z E do.

Like in 2.6 we obtained that s ( t ) is real analytic whenever s ( t ) >

- s - t , here we conclude in the same way that whenever

1 s,(t) > -say-t with s := - logd G,,t(a) = (1 - a ) s- ,

-t

all our theory is applicable to hence the function s,(t) is real

analytic and adjoint operator has eigenmeasure

with support dense in J F .

CHAPTER 2. NEGATIVE SPECTRA 72

Then for any a' E [a, a,,,] we have H := E Wl,, (a) for

p > 2 close to 2 and

implying for fixed z E 30 that

Thus ( f ix z E 30)

x (L;,,H) ( z ) =: r,(t)" .

This leads us to

Claim. Let a E [0, a,,,) and

t , := inf { t : s,(t) > -s,,-t) .

Then s,(t) is real analytic on the interval (t,, 0 ) and is equal (on this

interval) to s,, ( t ) for any a' E [a, amax].

Set S ( t ) := s,,,, ( t ) . Noticing that so ( t ) = s ( t ) , we conclude that

1) S ( t ) = s ( t ) for t E [to, 0 ) .

2) S ( t ) is real analytic whenever 5( t ) > -s,,,, ,-t.

So it remains to prove the following

Lemma. For any t the condition S(t) > -s,,,,, -t holds, i.e.

Proof. Throughout this proof for simplicity we denote L :=

and G := G,,,, ,t.

Function G (unlike G,,t for smaller a ) does not have zeroes at all

critical points of F , moreover for any z E 0 at least one of its preimages

is not a zero of G. Therefore we can consider the following operator on

probability measures on JF:

v I-+ Var (L*V)-' L*v ,

and by the Schauder theorem it fixes some measure which is an eigen-

measure of L*:

Clearly X 5 r,,,, (t).

Consider some neighborhood of the point a which is mapped by F

onto itself. Take small disk U inside it such that it's preimages Un

under the branches of F-" preserving a are disjoint.

Reasoning from [MS, 6.11 shows that suppv = JF thus v(U) > 0.

Since v is an eigenmeasure of L* and simple distortion estimates imply

we have

n

= A-" Gn (a) v(U) > ( ) .

Therefore

which implies G(a) < X and hence the desired inequality.

2.8. Phase transition and multipliers

We want to prove that the occurrence of a phase transition together

with (4) implies

Suppose it is not true and a phase transition occurs, though p(a) 5 p.

For any cycle b = {bl, ..bn) # {a) of order n (i.e. F n b l = bl) and

for N = k n we have

Thus

S(t) 2 -tlogdp 2 -tlogdp(a) = -ts- .

Together with real analyticity of S(t) this implies

But if a phase transition occurs at a point to then

S(to) = s(to) = -to% ,

and this condition is false.

We got a contradiction therefore iib) follows from the phase transi-

tion together with (4). H

2.9. Multifractal analysis

For simplicity we will work out the case of connected Julia set. We

already established the equality s(t) = P(t) - t + 1, and by [Mak2]

this implies

f (a) - 1 SUP = n(t) = s(t) = - P(t) = P(t) - t + 1 . a > O a log d

Moreover, since we proved nice convergence of the limits for p it will

imply the same for n (t) and f (a) .

It remains to analyze the Hausdorff spectra. Following the Subsec-

tion 2.7 we notice that for some (any) z E A(oo) we have

and following 1.4 this implies

dqt) - - dn(1-t) zl=l+d-n X lv'lt Iv - alK ,

where K, := -a,,,t > 0.

Hence the spectrum S(t) represents the @-spectrum with the neu-

tralized input from the point a. Particularly, that implies that if

w' := WLU is the restriction of w to some open set U not contain-

ing a, then

But (unlike the packing spectrum) the covering spectrum has "Haus-

dorff" properties, particularly if w = C w j , then c, (t) = supj cUj (t) .

Therefore we deduce that

To establish the inverse inequality it is sufficient to consider the

Hausdorff dimension spectrum f(a) and prove (cf. [Mak2, 2.21) that

for at := - l/Sr (t) we have the inequality

S(t) f (at) > inf (r + atS(r)) = t + atS(t) = t - - . 7 5' (t)

By the Subsection 2.2 for any t there exists a probability measure p =

,ut := ft ut with positive entropy such that

From the variational principleand convexity of g(t) we also deduce that

for any r < 0

hence, by the smoothness of g(r),

log d Z(t) = - / log ~ F ' d p ,

and also

logd s(t) = h, + t logd gt(t) .

Then, by Pesin's theory (see, e.g., result [Mafi3] of R. Maiie) for p-

almost every z the local behavior of w can be evaluated as

log w B6 log w Bg lim = lim lim log pBs 6-0 log6 6-0 10gll.B~ 6 3 0 log6

which together with the estimate of dimension (see [Mafi3] and [Y])

= (logd. S(t) - t logd gt(t)) (-logd gt(t))-l

gives the desired inequality !(at) 2 t - S ( t ) /S t ( t ) and completes the

proof. 4

3. Transfer operator on Sobolev spaces

3.1. Main theorem

We will consider functional spaces on the complex sphere, or some

open set 0 with

F-In c R ,

taken with spherical metric.

Transfer operator on the space of continuous functions is defined by

where preimages y of z are counted with multiplicities. Note that

where g n ( y ) = g ( y ) g ( F y ) . . . g ( ~ n - l ( y ) . Denote by X the spectral

radius:

n l / n X := lim llL l l m . 72-00

We assume that the weight g belongs to Sobolev space W17, for

some p > 2 close to 2, is non-negative and vanishes exactly at the

critical points of the dynamics F. Equivalently, we can assume that

g = h I F ' I ~ for small r > 0 and g, h, IF'IT E W17, for some (any)

number p > 2 sufficiently close to 2. Note that g, = h, 1 FA I T . We will assume that an additional condition is satisfied, namely

for some (any large) n, equivalently

const qn An > SUPS, ,

for any n and a constant q < 1.

Theorem E. Under the assumption (0) for p > 2 sufficiently close to

2 operator L acts on W 1 , and

ress (L, WI,,) < r (L, W17p) = .A ,

where r and re,, are spectral and essential spectral radii of L as an

operator on Wl ,, . Moreover, X is an isolated eigenvalue of multiplicity

one.

Proof. After some technical lemmas contained in the Subsection 3.2 we

will establish Ionescu-Tulcea and Marinescu inequality in the Subsec-

tion 3.3. As in the Subsection 1.8, together with finite rank approxi-

mation it will imply quasicompactness of the transfer operator.

CHAPTER 2. NEGATIVE SPECTRA 80

Following the Subsection 2.3, we obtain that X is an isolated eigen-

value of L, and also an eigenvalue of the adjoint operator L* with some

eigenmeasure v:

It remains to prove that suppv = J F , and then the argument of the

Subsection 2.5 will yield that multiplicity of X is one. But considera-

tions from [MS, 61 show that if this is not the case, then measure v

must be supported on a single fixed point or an orbit of order 2 (the

latter might happen only for Chebyshev polynomials). Let a be this

point (or one of these two points), then for even n we have Fn (a) = a

and hence

which contradicts the condition (0).

3.2. Technical lemmas

Lemma. There exists q l < 1 such that for small E > 0 we have

sup (g, 1 FA 1 -') < const (ql )n .

€ / t a u Proof. If we take E to be small enough so that q < (qX/ sup h) , then there exists ql < 1 such that

Therefore, noting that sup hn < (sup h)n we can write

< (sup h)n+ (const qn A ~ ) ?

proving the desired estimate.

2 Lemma. If 2 < p < i=-; a n d g , h, E WlYp, then LW1, c W I , ~ .

Proof. As before, it is sufficient to estimate

by const 1 1 f 1 1 Clearly

Here

and since we consider 2 < p < &, then r p + 2 - p > 0 and

The second term can be estimated by n

/ TP+~--P l l f l I ' ( S U P l ~ l j ( a h ) p d m 2

+ SUP(^)^ 1 (a ( I F / I ~ ) ) ~ IF^^^^+^-^ so it remains to show the convergence of the last integral, which, as

2 before, is implied by the condition p < .Ir; W

3.3. Ionescu-Tulcea and Marinescu

inequality

Lemma. For p > 2 close to 2, L acts on W 1 , and

IILZ"f Ill,, I const qn An llf lll,p + Cnllf llm 9

for some q < 1 and constant Cn.

Proof. Clearly

Choosing a small E, we can estimate the right hand side by

/ P('++-l) < const (ql)pn x~~ sup ( 1 F, 1 )

I const (ql)pn X P ~ Ilaf l l i 7

if we assume that ( p ~ + 2 - p) > 0, i.e. p is sufficiently close to 2.

Following the previous Subsection we obtain

which completes the proof.

Chapter 3

Parabolic case

In this chapter we will be concerned with parabolic polynomials, i.e.

those where all critical points are attracted either to (super)attractive

cycles or parabolic cycles.

In the first section we will establish the quasicompactness of the

transfer operator, and in the second apply it to the analyticity of spec-

trum, proving the following

Theorem B. B1. For a parabolic polynomial F, the function PF (t)

is real analytic on [0, HDim JF) and PF (t) = 0, t E [HDim JF , +oo).

B2. The derivative of PF(t) is discontinuous a t the point HDim JF if

and only if

L HDim JF > 2 - -

p + l '

where p is the maximal number of petals a t the parabolic points.

Dynamics in the parabolic case is not expanding, but expansive;

x 3 fi ' x R - x l dns =: ( S ) x d g

I(fi)S - (4SI

~ J O U SU!MOI~J ayl y w (x)3 JO aaredsqns P sre (x)dg a x d s ayl auyap arn 3 3 x JOJ -uo!?!ugaa

-[nav] u! paysyqre?sa sa!lxado~d :,!x)aur auros asn o s p I ~ M

arn ' ( [ ~ a ] aas) slas reqny papauuo:, J! papauuo:, L11~301 anrey 03 u ~ o u y

axre s~s~uroulC~od y:,ns 'l~red't! JBJ 08 L~~renluana sluurod OM) Lure 'L~aurreu

111. There exists constant C such that any x, y E U can be joined

inside U by the arc of length less than CI x - y 1 .

Remark. The third property implies that for any 6 > 0, f E C(U)

Idea of the Proof. As in [ D H , X.2.41, one can take some neighborhood

of the Julia set and then cut out a few angles, inside which critical

points approach parabolic cycles. H

1.2. Transfer operat or

For z E U and t E R we define

Clearly for bounded subsets X c C for any n gn E Lip(X). This

allows us to consider the transfer operator on the Lip(U):

where preimages y of z are counted with multiplicities. Note that

Definition. For t E R and positive integer n we define the partition

function

Denote also

X* := limsup v w . n--+w

Theorem. Fix t R.

I. There exist a probability measure v := vt on JF, and a positive

number X := At such that

L*v = Xu, suppv = JF ,

and for any positive integer n and z E U

thus X is the spectral radius of the operator Lg in C(JF).

then X is the spectral radius and an isolated eigenvalue ofmulti-

plicity one of the operator Lt : Lip(U) + Lip(U). Furthermore

for any positive integer n and z E JF

The first part of the theorem is proved in the sections 1.3-1.4 below,

and the second in the sections 1.5-1.9. Throughout these sections we

often omit t , denoting vt by v, At by A, etc.

1.3. Existence of conformal measure

Clearly, for p E Prob(JF), L*p is a positive non-zero measure. Thus

we can consider the following operator on Prob:

P : p I-+ L*P

VarL*p '

It fixes some measure v which is the desired one. Condition suppv = JF

holds since z E suppv implies F-n(z) c suppv and preimages of

any point z E JF are dense in JF.

1.4. Partition function

Lemma. For any positive integer n and x, y E U

Proof. Since U is compact it is sufficient to prove this for x, y E V,

where V is some small connected subset of U . Denote by %k compo-

nents of F - ~ v . By the Proposition 1.1. we know that

supdiam %k = o(l) , k --+ oo , j

thus SUPy 9

W k := sup - - ,o(n)

j i n f v g

because g is separated from zero on U . Therefore

This implies our Lemma.

Remark. We also proved that for x, y in one component of connectivity

Now we can prove (2 ) . In fact for any x E JF we have

So ( 2 ) holds and the spectral radius of L, in C ( J F ) is equal to

CHAPTER 3. PARABOLIC CASE

We turn now to the proof of the second part of Theorem 1.1.

1.5. Prevalence of the expanding branches

Lemma. Fix Q > 1. For z E U denote

Then

Also

inf l(Fn)' (z) l 2 e -o(n) . ZEU

Proof. Take V to be a neighborhood of z in U. Then

hence by the first part of the Theorem 1.2

which implies

The second statement can be proved in the same way.

1.7. Ionescu-Tulcea and Marinescu inequality

Suppose the condition (3) is satisfied. Normalize g ( z ) by setting

Denote

then

Lemma. In the above notation there exist 0 < q < 1 and Mn = eo@)

such that

Remark. Note that in our construction, we obtain

4 " eO(l) ((Q 2+t 1 ) + Q - ~ ) lln

Choosing Q properly and taking large n, we can obtain some value of

q , depending only on X and t . This method, applied in parabolic or

hyperbolic case, will give an estimate on the radius of essential spectrum

of the transfer operator Lt , in terms of t , P( t ) , and P ( f oo) only.

CHAPTER 3. PARABOLIC CASE 92

However, this does not provide us with an estimate for the absolute

value of the second eigenvalue.

Proof. First we want to estimate

:= # N n ( z ) inf I (F")' (z) l SUP hn(y) + IILnlll, Q-n . zEU yen/, ( z )

If t 2 0 then for h ( z )

and

< eo(n)pn SUP hn(y) 5 llhnll, - Y ENn ( z )

Choose Q > 1 such that pQ2 < 1. Then

If t < 0 then

Clearly Lemma 1.5 implies that X > 1, so we can choose Q > 1 such

that inequality Q2+t / X < 1 holds. Then

In any case we can pick large n (denote it by N) such that

for some q (C is the constant from the Proposition 1.1).

For f E Lip consider such continuation f E C(U) that

for example it is true for

Now if points z , z' E U are sufficiently close we have

where we denote by y' the preimage of z' obtained by the branch of

F-" for which F-"(2) = y. Using (6 ) we can estimate A by

and B (again, we suppose z and z' are sufficiently close) by

2 sup Ix-y1<6 l x - 91

Therefore

with

Now by induction we have

with Ck Too,

This implies (5).

CHAPTER 3. PARABOLIC CASE

1.8. Quasicompactness

A Banach space linear operator L is called quasicompact if

for some compact operator K and some integer N. An equivalent state-

ment is that the image [ L ~ ] of L~ in the Calkin algebra (= bounded

operators modulo compact) has norm strictly less than one. This im-

plies that the spectral radius of [L] in the Calkin algebra is strictly

less than one, and therefore there is r < 1 such that the part of the

spectrum of L lying outside of the disk {lzl 5 r ) consists of a finite

number of eigenvalues that all have finite geometric multiplicity.

Lemma. Any operator L on Lip satisfying ( 5 ) is quasicompact.

Proof. Cover J by some equilateral triangle B. By the Whitney exten-

sion theorem there exists bounded operator

W : Lip (JF) + Lip (B) ,

with (Wf) IJ, = JF.

Set K := IIWIILi, 2 1 and fix N such that

1 IILNf l L i p 5 llf llLip + M If lloo, f E Lip

Also fix small E (what we need is M K E 5 A) and take partition of B

into equilateral triangles with sides less then E . Define an operator P

on Lip(B) by the requirement

P f = f on the vertices of our small triangles, P f (x) is a linear function of Rex , Im x inside any of them.

Then d i m P ( ~ i ~ ( ~ ) ) < 00, and P < 2. 1 L i p ( B )

Set now P := PW, P : Lip + Lip. It is clear that dim P (Lip) < oo

thus to prove that L is quasicompact it remains to check that

since llPllLip - < 1 1 ' w L i p 5 2 ~ 7 and Lip

we have

Returning to the proof of the second part of Theorem 1.2, we see

that the last two lemmas imply that 1 is the spectral radius and an

isolated eigenvalue of L =: Lip 6, therefore X is the spectral radius and

an isolated eigenvalue of L, : Lip 6 . It remains to show that

dim n ker(Lg - A)" = 1 . n > l

1.9. Multiplicity

In this section we denote by L the unnormalized operator L,. First

we prove

dim ker(L - A) = 1. (8)

Observe that if f satisfies Lf = Xf with X > 0 then Llfl = Xlfl.

Indeed,

(Xlf 17) = (ILf 17) I (Llf 1 7 4

= (If l,L*v) = (If I , Xv)

and Ll f 1 = XI f 1 v-a.e. and hence everywhere because suppv = J F .

Next observe that if f is a real-valued eigenfunction, L f = X f , then

either f 2 0 or f I 0. Otherwise, we can find a nontrivial eigenfunction

f 2 0 (e.g. f = I f 1 - f ) which is zero at some point. The equation

then implies that f must vanish on a dense set, hence everywhere.

Note that we also proved that eigenfunction f is strictly positive.

Thus equality

implies (4): in fact, for any n and z E JF

Therefore we also have

Suppose now that we have two real functions fl, f2 satisfying

Normalizing them by the condition

we have

and fl = f2. This proves (8).

It remains to show that

ker(L - = ker(L - A).

CHAPTER 3. PARABOLIC CASE

Suppose this is not true. Then there are f # 0 and h such that

L f = X f , Lh= Xh+cf, ( c # 0).

By induction, we have

Lnh = An (h+ncX-'f)

which contradicts the relation (9).

2. Analyticity

2.1. Analyticity

Define P ( t ) := log At. Area estimate in the Subsection 1.5 shows

that P ( t ) 5 0 for t 2 2, hence A+ 5 1. On the other hand, existence

of the parabolic cycle with multiplier 1 gives us A+ 2 1. Thus A+ = 1,

and we can reformulate Theorem 1.2 in the following form:

As long as P ( t ) is positive, At is the spectral radius and an isolated

eigenvalue of multiplicity one of the operator Lt : Lip(U) + Lip(U).

Let 6 be the first root of P( t ) . Then At for t < 6 is an isolated

eigenvalue of multiplicity one of the operator Lt which depends real-

analytically on t , hence, by perturbation theory, the pressure P ( t ) is

real analytical on this interval. On the other hand, P( t ) is decreasing

and non-negative, thus P ( t ) = 0 for t > 6.

To complete the proof of the Theorem B it remains to analyze the

phase transition at the point 6.

2.2. Phase transition

By the paper [DU2] the Bowen's formula holds for parabolic maps,

hence 6 = HDim JF . By [ADU] (see especially Theorem 9.9) there

is a 6-conformal measure v = vb (which is either Hausdorff or Packing,

depending on the dimension), and it admits an equivalent (a-finite)

invariant measure p, which is finite if and only if the inequality (4)

holds.

Now, if (4) holds, such a measure p is finite, thus (being equiva-

lent to the conformal measure v) it maximizes the expression in the

variational principle:

and has a positive entropy, hence (by the variational principle)

The last integral is positive (it is equal to h,/S), therefore Pt(S-) <

0 = Pt(S+) and the derivative of the pressure is discontinuous at S.

On the other hand, suppose that Pt(S-) < 0. Then for each t < S

there is an invariant measure pt (equivalent to the (A t , t)-conformal

measure vt: particularly, pt = ftvt, where ft is the eigenfunction

of the transfer operator), for which S log lFtldpt > -Pt(6-) > 0.

Considering the limit of ft as t -+ 6 (and normalizing them properly)

it is easy to construct a finite invariant measure p equivalent to v = us

and therefore deduce the inequality (4).

Chapter 4

Semihyperbolic case

In this chapter we restrict ourselves to the study of non-recurrent

2 quadratic polynomial F (z) = z + C:

We prove the following

Theorem D. For a non-recurrent quadratic polynomial F (2) = z2 + c

either

(i) PF (t) is real analytic on [0, +oo), or

(ii) A phase transition occurs: there exists to > HDim JF such that

PF (t) is real analytic on [0, to), and 1 PF (t) = - - 2 t log (lim inf,,, 1 (F")'(c) 1 'In) on [to, +m) .

The chapter is organized as follows: in the Section 1 we introduce

Hofbauer tower, changing the dynamics and forcing expansion. In the

Section 2 we prove quasicompactness of the transfer operator, and in

the Section 3 investigate its eigenvalues and their connection to spec-

trum,

1. Yoccoz puzzle and Hofbauer tower

1.1. Preliminaries

Define

1 s(t) := lim sup - logz

n+oo n C I ( ~ ~ 9 ) ' l -t 7

y € F - * ( z )

for some (any - there is no difference, see Proof of the Lemma) z E

Lemma. For any z E A(m)

Therefore

Proof. Without loss of generality cp conjugates F with dynamics T :

z H z2 on ID-:

F o p = p o T .

Differentiating the identity Fn o cp = cp o Tn, we obtain

Applying this equality to the preimages c E T-* of some fixed point

E E ]ID- we notice that the right side is =: 2n, thus (taking to power t)

where y = p(C) is a corresponding preimage of z = p(<) under F n .

The points 5 are equidistributed on the circle

with (r, - 1) =: - 2" • Therefore summing over all C E T-n< ( y E

F-%), we have

which completes the proof.

1.2. Tower construction

For non-recurrent polynomials dynamics on the orbit of the critical

point is expanding, i.e

Q, := liminf IF"'(c)~' > 1 . n+oo

CHAPTER 4. SEMIHYPERBOLIC CASE 105

Consider Yoccoz puzzle (see [Mi21 and [Hu] for the construction and

its properties) and denote by Pn (0) the puzzle-piece of depth n + no

containing 0 (in this case 0 cannot lie on the boundary of a puzzle-

piece). Since for such Julia sets diameter of puzzle-pieces tends to 0

when depth increases, we can choose no in such a way that Po(0) is

disjoint from wO.

Denote by 0 formal disjoint union of all puzzle pieces of depth no + n l , n l to be chosen later. Our tower will be a union of floors equivalent

to subsets of a:

where Fkr c 0. Define floors by

1.3. Metrics on the tower

We define metrics on the tower by setting

Q~- (~ ' - ' ) Izl - z21, k1 = k2 > 1

Izl - kj = 1, Zj are in p ((a, k l ) , ( ~ 2 , k2)) =

the same puzzle-piece

( oo otherwise , 1

where Qp is some constant satisfying Q,z > Qp, to be chosen later. To

simplify notation we will write Ix - yl instead of p (x, y) for x, y E I.

1.4. Dynamics on the tower

Set

uk' := Fk+l' , z ) ~ ' := Fk ' \Ukr ,

and

u := ( k ) , Dk := (Z)kl,k) .

Define F by

Note that F is undefined for some points on the first level of the tower,

however transfer operator and its' formal adjoint are still well-defined

(we adopt an agreement that sum of empty set of terms is zero). To

simplify the formulas we will write F, in place of n-th iterate pn.

1.5. Properties of the tower

Proposition.

i] Dynamics F maps 1-to-l

Uk to Fk+1 ,

Vk, k 2 2 to a union of some components of Fl .

ii] If x, y are in one component, i.e. Ix - yl < oo, then they have

preimages in the same components.

iii]

iv] By the choice of nl we can make expanding, i. e. for some C

and Q > 1

IFAI > CQn .

v] For some Q > 1 for any n and C we can choose T in such a

way that y E .Fk, k 2 T which after less than n iterations goes

down

i],ii]. Follow directly from the construction. The property ii] means

that we can estimate IV (Lf (z))l through estimating IVf (y)l for y E

F-l.

iii]. First note that

6, := diam (F (Pk(0))) x IF;(C)~-' .

Take y E Pk \ Pk+1 = Dk , k 2 n1, from the distortion theorems

~ ~ ' ( 9 ) 1 = I F ~ ( Y ) I F ( ~ - ~ ) ' F ( Y ) 1 = IyI I F ( ~ - ' ) ' ( C ) ~ = diam ( F ( P ~ (o)))$ 1 F ( ~ ~ ) ' ( c ) I

1

n / F ( k l ) ' ( C ) T .

Therefore

~ ' ( y , k ) l = Q , - ( " ~ ) I F ~ ' ( ~ ) ~ 1 -

- 1 F ( k - l ) ' ( C ) 2 k

= Q~ 1 1 2 Q ~ - ~ Q ~ - T

1

iv], v]. Follow from Qp < Q, 1 .

2. Analysis of the transfer operator

2.1. Transfer operator on the tower

We will consider two function spaces on the tower: C (I), Lip (7)

with norms

f = f ) := SuPllfllW,Fk k

For t E R we define operator Lt on C ( I ) by

note that

Denote by At the spectral radius r ( L t , C ( I)) .

Lemma. Lt is bounded operator on C ( I ) , and At 2 2S(t) .

Proof. The boundness of L f follows from the property v] of the tower.

To prove the inequality take some point z E A(m) such that

For such z and n 2 nl all points y E F-"z satisfy pn ( y , 1 ) = ( z , ~ )

and F~~ ( y , 1 ) = ~ ~ ' y . Thus by Lemma 1.1 for n 2 nl

2.2. Ionescu-Tulcea and Marinescu inequality

Lemma. Suppose At > Q,-'~. Then for n >> 1 and T >> n we have

Proof.

For z E F k , k < n we will use the estimate (since all functions are

Lipschitz the gradients are defined almost everywhere, and later we will

be able to consider an essential supremum)

ye@-" (t)

5 sup Lip (f, .Fk) Q-n A; eo(n) + C I Ln f ( z ) I , k

where C is an absolute constant.

Note that z E &, k 2 n has only one preimage y E Fk-, and in its

neighborhood Fn increases all distances exactly by a multiplier of Qpn,

thus

Using the latter inequality for z E Fk, k 2 n we obtain that for

n >> 1 for sufficiently large T >> n

sup Lip (Ln f ) = sup ess-supFk VLn f k k

< ' A; Lip ( f ) + C IILnf 11, - 5

Now estimate in the same way 1 f (z) 1 . -tn An. Then for Since At > Q,-~, for large n we have Q p < 5(C+1)

z E Fk, k 2 n we can estimate

where y E Fn-k is the only preimage of z.

For z E Fk, k < n number of preimages of z on each level of the

tower is bounded by some constant K (depending only on n). We can

CHAPTER 4. SEMIHYPERBOLIC CASE 112

choose T in such a way that for some q < 1 and any y E Fk, k 2 T

which after less then n iterations goes down

Then

Together these estimates give us

CHAPTER 4. SEMIHYPERBOLIC CASE

Therefore

1 < - A," supLip(f) + ( C + l ) IILnfll, - 5 '

1 < - A," sup Lip (f) - 5 '

2.3. Finite rank approximation

Lemma. For any integer T and 6 > 0 there exists a finite rank oper-

ator M in Lip (7) such that

IIMIILip 5 1 7

I f , 5 6 IIfllLip,Fk 7 I ~ < T ,

Mf(z) = 0 , k > T .

Proof. Cover .Fk , k < T with a grid of triangles q5 of size 6 << 1. Define

M f as a continuous function vanishing on F', k 2 T and satisfying

f at all vertices M f = { is linear in each triangle q5 ,

on F', k < T. The properties are obvious.

CHAPTER 4. SEMIHYPERBOLIC CASE

2.4. Quasicompactness

Claim. If At > Q ~ - ~ then

r(Lt, Lip (7)) = A t ,

and

Proof.

First recall that

r (Lt , C (7)) = At

Also

and At 5 r (Lt , Lip (7))

To prove the converse, by the Ionescu-Tulcea and Marinescu inequal-

ity choose E > 0 and N such that

IILNf l l L i p 5 'F-' llf llLip + K llf ll,

Define inductively an increasing sequence { M k ) by

Then

and we can choose a constant C satisfying

c A,"-& + 1 5 C A ; and K 5 C M l .

By induction,

1ILNkffIlLiP 5 ',"k-ek l l f llLip + ' M k l l f l l ,

< ' t N k llf llLip 7 N

therefore

r (Lt , Lip (l)) I At ,

which implies the desired inequality and thus the first statement.

To prove the second statement choose n and T such that

and a finite rank approximation M for S < & Xr .

Then

and

3. Analyticity of spectrum In the subsections 3.1 - 3.3 we will assume that At > Q ~ - , , i.e.

Claim 2.4 holds.

3.1. Eigenvalues and multiplicity

Fix t and denote X = At := r (Lt, C ( I ) ) .

Lemma. ker (Lt - A) # (0). Moreover there is a strictly positive

eigenfunction f = f t .

Proof. See [Ru6, Theorem 2.21.

Idea of the proof: since re,, (L t , Lip ( I ) ) < r (L t , Lip ( I ) ) = A,

we have some eigenvalues X j of absolute value A, and a decomposition

where Xo is the subspace corresponding to the rest of the spectrum.

Then

Here LrXo = o ( A n ) and L r X j = X y X j , implying that one of the

X j ' s is positive and hence is equal to X = r (L t , C ( I)) .

Also it follows that there is non-negative (and hence strictly positive:

if f ( z ) = 0 then f ( y ) = 0 for Fny = z , from continuity f s 0 on JF

and by Sublemma everywhere) eigenfunction f with eigenvalue A.

Sublemma. For f E Lip (7)

Lt f = X f implies f = 0 .

Proof of Sublemma. For fixed z E 7 and y E distortion estimates

imply

l ~ : ( ~ ) l - ' x dist ( y , j F ) .

Therefore

< X-" N IF;(Y)I-~ (y , jF )

YEP-" ( z )

< X-" N c I ~ A ( Y ) I - ~ - ~

y€F-" ( z )

1 < A-" - N

CQn ~ ~ ; ( y ) l - ~ 5 e O ( " ) / Q - " - 0 .

72-00

y € F - n ( z )

Consequently f = 0 and Sublemma is proved.

3.2. Conformal measures

Consider the formal adjoint L* to L, acting on the space of measures

Lemma. There is a probability measure v = vt supported on 7 with

L*v = X v .

Proof. Clearly

CHAPTER 4. SEMIHYPERBOLIC CASE 119

Take some positive measure p supported on the first floor of the tower

and define

Then

thus to find v it is sufficient to prove that some subsequence {p,) has

a weak limit. The only trouble is that 7 is not compact.

But one can easily check by induction that for small Q > 1 and large

positive C

which is sufficient.

Remark. Denote SF := Uk { ( z , k ) E Fk : z E J F ) . Then jF is invari-

ant under F and suppv > SF.

3.3. Pressure and spectrum

Lemma. We have X = 2'@).

t Proof. We know that X 2 2S(t) 2 Q,-5. Assume the claim of the

Lemma is not true. Then X > 2S(t) . Chose such X1 and Qp that

By the definition of s ( t ) the series

is convergent, hence we can define a finite measure uo on JF by

Note that duo(Fy) = X1 ~ F ' ~ l - ~ d u ~ ( y ) .

Now transfer this measure to the tower by setting

we obtain a finite measure again (since X y QMnt < 1). It is easy to

check that the mentioned property of uo implies

and hence L* ul = X1 u1. Therefore for the positive eigenfunction f we

can write

Since f is strictly positive this implies X = X I , and we obtain a contra-

diction. rn

Claim. We have dim ker (Lt - x ) ~ = 1, i.e. X is a simple eigenvalue.

Proof. If we consider only functions in C JF , the same reasoning as ( - 1 in [MS, 83.61 gives us that dim ker (Lt - A) = 1, and then Sublemma

implies this for f E Lip ( 7 ) .

Lemma 3.1 shows that we have a non-negative eigenfunction f :

(L, - A) f = 0 .

If X is not a simple eigenvalue, then there exists function h such that

(Lt - A) h = f .

Therefore

Thus v-a.e. we have f = 0, hence everywhere on jF and (by the

Sublemma) in 7.

3.4. Analyticity

Claims 2.4 and 3.3 show that if for some t' we can choose Qp in

such a way that At > QPwt > Q,-5 then 2S(t) is a simple isolated

eigenvalue of Lt : Lip (7) 0. Clearly it implies that Lt depends real

CHAPTER 4. SEMIHYPERBOLIC CASE 122

analytically on t therefore s(t) and hence P(t) is real analytic in the

neighborhood of t' (see [MS, 4.11).

By Lemma 2.1 At 2 2'@), thus inequality 2S(t) > Q,-g is sufficient

to find Qp satisfying our conditions. On the other hand we always have

2s(t) 2 QC-i thus to prove the main Theorem we only have to check

First note that

6, := diam (F (Pn(0))) x I FA(C) 1-l .

We can choose a point z E A(oo) in each part of R, so that for any

n one of them has a preimage y E Pn \ Pn+l under the F-". From the

distortion theorems

x diam (F (pn (0))) 1 F("-')'(c) 1

Then for the corresponding z:

and taking limsup as n -+ oo we obtain the desired inequality.

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